Loud noise-No CEL or other dash lights

My 04 Prius has a loud screaming noise coming from the engine bay when I start it up and when I shut it down. I have no CEL light or other warning lights like ABS. No symptoms like brake loss or anything, it drives completely normal and this noise has been going on for about 10 months now. I took it to my two local Toyota’s and they did a full inspection and couldn’t find anything. My heater and AC work fine, I know what the ABS pump sounds like and this is different. Very loud and makes heads turn LOL. Does not happen every single time, but it’s like 90% of the time when I go to turn it on it’ll scream for like 5 seconds then the engine will start up, when I push the button to turn it off it’ll wait 2 seconds then it’ll scream for 5-10 seconds. I’m going to attach a video and hopefully someone out there knows what this is, thank you!

 

Which pump? And wouldn’t I get any lights on the dash? & I know, not sure if they half a$$’d the inspection or what but I have paperwork to prove it.

 

So not the storage tank pump? Would you happen to have the part number or link, I couldn’t find anything on google that says coolant heat recovery system pump, thank you!

 

Thank you! The noise is pretty annoying but I can live with it for now, thanks again for your help, I had no clue what to look for.